Gives a deeper thump. It's an art form, getting it right... something I've never been able to master.
Closest I ever came was using Teddy Bear... that's right, a large Teddy Bear... he worked well, gave me a nice deep thump. He got stolen. Since then I've tried, I guess, every way one can dampen a drum.... from pillow, blankets, to the more store bought stuff... attachable dampeners, foam strip added and sometimes included on the drumhead... currently I use a device that glues to the bottom of your bass drum and is adjustable, using velcro, so you can loosen and tighten the 'pillow' that sits against the beater head. Well.. actually no...
currently I use an electronic kit so I have no need of any dampener.